Let V be an inner product space. Let be a complete orthonormal sequence in V. Suppose that with and . Prove that .

My proof so far:

I definitely don't think I can pull that sum out in the first step because the has the sum with it as well. I was thinking of trying the following:

I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.